Question: Multiply the following complex numbers, marked as blue dots on the graph: $[2(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))] \cdot [2(\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi))]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $2(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))$ ) has angle $\frac{1}{4}\pi$ and radius $2$ The second number ( $2(\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi))$ ) has angle $\frac{5}{3}\pi$ and radius $2$ The radius of the result will be $2 \cdot 2$ , which is $4$ The angle of the result is $\frac{1}{4}\pi + \frac{5}{3}\pi = \frac{23}{12}\pi$ The radius of the result is $4$ and the angle of the result is $\frac{23}{12}\pi$.